Theorems · Theorem · category theory

CategoryTheory.LocalizerMorphism.commShift_iso_hom_app_assoc

∀ {C₁ : Type u_1} {C₂ : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C₁]
  [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂] {W₁ : CategoryTheory.MorphismProperty C₁}
  {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) {M : Type u_3}
  [inst_2 : AddMonoid M] [inst_3 : CategoryTheory.HasShift C₁ M] [inst_4 : CategoryTheory.HasShift C₂ M]
  [inst_5 : Φ.functor.CommShift M] {D₁ : Type u_4} {D₂ : Type u_5} [inst_6 : CategoryTheory.Category.{v_3, u_4} D₁]
  [inst_7 : CategoryTheory.Category.{v_4, u_5} D₂] (L₁ : CategoryTheory.Functor C₁ D₁) [inst_8 : L₁.IsLocalization W₁]
  (L₂ : CategoryTheory.Functor C₂ D₂) [inst_9 : CategoryTheory.HasShift D₁ M] [inst_10 : CategoryTheory.HasShift D₂ M]
  [inst_11 : L₁.CommShift M] [inst_12 : L₂.CommShift M] (G : CategoryTheory.Functor D₁ D₂)
  (e : Φ.functor.comp L₂ ≅ L₁.comp G) (m : M) (X : C₁) {Z : D₂}
  (h : (CategoryTheory.shiftFunctor D₂ m).obj (G.obj (L₁.obj X)) ⟶ Z),
  CategoryTheory.CategoryStruct.comp ((CategoryTheory.Functor.commShiftIso G m).hom.app (L₁.obj X)) h =
    CategoryTheory.CategoryStruct.comp (G.map ((CategoryTheory.Functor.commShiftIso L₁ m).inv.app X))
      (CategoryTheory.CategoryStruct.comp (e.inv.app ((CategoryTheory.shiftFunctor C₁ m).obj X))
        (CategoryTheory.CategoryStruct.comp (L₂.map ((CategoryTheory.Functor.commShiftIso Φ.functor m).hom.app X))
          (CategoryTheory.CategoryStruct.comp ((CategoryTheory.Functor.commShiftIso L₂ m).hom.app (Φ.functor.obj X))
            (CategoryTheory.CategoryStruct.comp ((CategoryTheory.shiftFunctor D₂ m).map (e.hom.app X)) h))))

Defined in

Mathlib.CategoryTheory.Shift.Localization

Cited by

0 results in Mathlib

Foundations

Depth 89 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryAddMonoidCategoryTheory.HasShiftCategoryTheory.HasShiftCategoryTheory.Functor.CommShiftCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsLocalizationCategoryTheory.HasShiftCategoryTheory.HasShiftCategoryTheory.Functor.CommShiftCategoryTheory.Functor.CommShift